Genesis 17:15-17 meaning

God blesses Sarai by changing her name, Sarah shall be her name (v 15). Sarah became the "princess" or "woman of nobility" of the covenant to Abraham. Both names mean "princess." However, now Sarah would be a princess of God. The change in name indicated the multitude of people who would come forth from her; many kings, nations, and leaders would come from her children. Sarah was the mother of the son of the promise, Isaac, by a special act of God (Genesis 17:19-22, 18:7-15, 21:1-6). Isaiah called her the mother who gave birth to Zion, which means Israel (Genesis 51:2). 

God's second blessing to Sarah is the announcement that He will give you a son by her (v 16). This is the first time that God says Abraham will have a son by Sarah. God had promised a son before but never said that Sarah would be the mother. Abraham was given assurance by God that Sarah would be the mother of the son He had promised. This is significant because of Sarah's age. Another blessing to Sarah is God declaring that she will become a mother of nations and that kings of peoples will come from her (v 16). 

Then Abraham fell on his face and laughed (v 17). One cannot know exactly what kind of laugh this was; perhaps a simple smile, chuckle, or a loud outburst. The context shows it expressed how incredible it was for Sarah to have a child due to her age. The next passage shows that Abraham had come to believe the promised line would come through Ishmael.